### bricks in the wall

much of the point here is to “explain why”

the figure in the upper right… “pascal’s triangle”…

works the way it does: students gladly learn

in minutes how to write out rows by adding

pairs taken from previous rows (according to

the formula in the lower left, though they

know it not; eventually they’ll have to be

encouraged to care but of course one need

not speak of thorny notational issues at

every single opportunity; when the student

is ready the code’ll have been there waiting

all along). the thing is: what the heck does

this ritual-adding-together have to do

with “counting subsets”, then, eh? because

when we “use” these numbers in our little

probability-and-counting problems back

in class, we’ll be referring to “N choose R”

all the time: the *number of ways to
“choose” a set of R things from a set of N*.

well, put like that…

the thing is more or less obviously

then to look carefully at arrangements-of-subsets

and see if we can piece out

“what comes from where”.

my best shot so far.

better with the handwaving of course.

May 7, 2010 at 2:18 pm

Due to its simple construction by factorials, a very basic representation of Pascal’s triangle in terms of the matrix exponential can be given: Pascal’s triangle is the exponential of the matrix which has the sequence 1, 2, 3, 4, … on its subdiagonal and zero everywhere else.

good ol’ wikipedia.

May 7, 2010 at 2:32 pm

okay suppose you’ve got to know about N

but all you know about is N-1. somebody

gives you N things anyhow and sez

“how many ways can i get R things

out of these here?”.

take one of the N things and paint it blue.

now there’s the N-1 colorless things and the blue one.

we’re supposed to pick a set of R things somehow.

okay… i say now there are *two* ways i might

go about getting such a set:

pick the *whole set of R things*

from the nonblue N-1…

there are {N-1 \choose R} ways to do this…

or

pick *only R-1 nonblue things* from

this same N-1, plus the blue thing for

the full quota of R things altogether:

there are {N-1 \choose R-1} ways

to do this.

so we’ll’ve seen that

.

(not the same formula as in

my handwritten display but

the same fact.)

i forget who put it to me like this

but this is the one that stuck.

May 7, 2010 at 7:39 pm

ktm: dy/dan @ ted (etc.)

http://kitchentablemath.blogspot.com/2010/05/dan-meyer-at-ted-x-nyed-blow-up-us.html

May 7, 2010 at 8:01 pm

back on the message of the day.

i did a new issue of the

“lectures without words”

series today: MathEdZine 0.8

(the K_n — K_4 remix).

it’s a classic “minicomic” format:

copy on both sides of a sheet

of typing paper, fold the long way

first, then fold again the short way,

staple along the spine (with an

ordinary desk stapler this will

entail folding over a few pages;

with a “longarm” stapler one

can just shove it it and bang down),

and trim it at the top. voila:

an eight-pager just like cynicalman.

without the brilliant art and

longpracticed production values.

anyhow, it’s the first zine i’ve made

with this particular “imposition” and

so i get a tiny bit of a thrill from it.

there’s also a sort of, oh-heck-give-up

to the whole thing though since i’d

never even thought of *doing* a remix

if the “unstaplebook” format of

the original 0.6 and 0.7 wasn’t

distracting almost every subject

i’ve had a chance to work with…

i’m glad i *did* the unstaplebooks.

part of the point *is* the very

what-the-heck-is-*this*

effect that the very format

of the document inspires…

among other things, this gives the “teacher”

(that would be me in the interviews i’ve had

but is always somebody else in my imagination)

a chance to bail out on the math

(i’ve-never-been-good-in-math

is just about as reliable a response

to being confronted with a MEdZ

as what-the-heck-is-this is) and

talk about *publishing* and, more

to the point for “educators” i suppose,

*self*-publishing and the DIY vibe

generally. teach a kid to write up

cool math facts and they’ll fish

for the rest of their life kinda thing.

but i’m laying off of ’em

(“unstaplebook” micro- and nano-

-zines) once i’ve filled in the

and

issues at 0.4 and 0.5

i think.

May 7, 2010 at 8:05 pm

the display in the photo

appears in both K_n

and the “remix” issue.

i meant to’ve said.

(why doesn’t somebody

just *interrupt* me

when i get going like that.)

May 8, 2010 at 3:46 am

I’m going to try to respond to kate(t)’s challenge by using binomial expansion to create pascal’s triangle. Maybe.

1

x | y

first put xs on the left: xx xy, then put ys on the left: yx yy

xx | xy yx | yy

first put xs on the left: xxx xxy xyx xyy, then put ys on the left: yxx yxy yyx yyy

xxx | xxy xyx yxx | xyy yxy yyx | yyy

counting, we get 1 2 1 0 + 0 1 2 1

like, no surprise, multiplication by 11.

Need to play and clean, and develop cooler graphics than I’ve ever done.

Jonathan

May 8, 2010 at 3:57 pm

it looks to me like “pascal’s triangle”

… why do the binomial coefficients

behave the way they do…

is the heart of the matter for

kate’s problem. everything else

is easier for *me* to see, anyhow.

i’ve *also* got notes on how to

lay out answers and work ’em

by rote. i drilled several classes

in expanding (P + Q)^N

for probability-and-stix

applications for quite a bit

more than the standard presentations

would ever dream of doing just

so i’d feel like they’d learned

to do *something* calling for

honest algebraic manipulation

of binomial coefficients.

i had to cut out some of the

“pretend to learn a bunch of

technical terms for regurgitation”

stuff to do it though. oh the horror.

i’m unlikely to submit anything myself.

i’ve finally given in to the need to

develop cooler graphics than i’ve ever done.

with ink and paper and whatnot though…

making real objects in the real world.

computers are just way too expensive

and frustrating without massive

institutional support.

i can’t work this damn copier and

it breaks my heart.